Scalars and Vectors

This lesson covers the difference between scalar and vector quantities — a fundamental distinction in physics — as required by the Edexcel GCSE Physics specification (1PH0), Topic 1: Key Concepts of Physics. You need to be able to classify quantities as scalar or vector, represent vectors using arrows, and add vectors using scale diagrams.

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What Is a Scalar Quantity?

A scalar quantity has magnitude (size) only. It does not have a direction. When you state a scalar, you give a number and a unit — nothing more.

Examples of Scalar Quantities

Quantity	Unit	Example
Speed	m/s	A car travelling at 30 m/s
Distance	m	The track is 400 m long
Mass	kg	The ball has a mass of 2 kg
Temperature	°C or K	The water is at 100 °C
Energy	J	The battery stores 5000 J
Time	s	The race took 12 s

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What Is a Vector Quantity?

A vector quantity has both magnitude (size) and direction. When you state a vector, you must give a number, a unit, and a direction.

Examples of Vector Quantities

Quantity	Unit	Example
Velocity	m/s	A car travelling at 30 m/s due north
Displacement	m	400 m to the east
Force	N	A 50 N force acting downward
Acceleration	m/s²	9.8 m/s² downward
Momentum	kg m/s	600 kg m/s to the right

Exam Tip: A very common exam question asks you to classify quantities as scalar or vector. Learn the examples above. Remember: speed is scalar but velocity is vector; distance is scalar but displacement is vector. The vector version always includes a direction.

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Scalar vs Vector: Key Differences

Feature	Scalar	Vector
Magnitude	Yes	Yes
Direction	No	Yes
Can be negative	No (magnitude is always positive)	Yes (direction can be opposite)
Addition	Simple arithmetic	Must consider direction

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Distance vs Displacement

These two terms are often confused but have very different meanings in physics:

• Distance is a scalar — it is the total length of the path travelled, regardless of direction. It is always positive.
• Displacement is a vector — it is the straight-line distance and direction from the starting point to the finishing point.

Example

A runner completes one lap of a 400 m track, returning to the starting point.

• Distance = 400 m (the total path length)
• Displacement = 0 m (they are back where they started — there is no straight-line distance from start to finish)

Exam Tip: If an object returns to its starting point, its displacement is zero but its distance is not zero. This is a favourite exam question.

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Speed vs Velocity

Similarly:

• Speed is a scalar — it is the rate of change of distance. Speed = distance ÷ time.
• Velocity is a vector — it is the rate of change of displacement. Velocity = displacement ÷ time.

An object can have a constant speed but a changing velocity — for example, an object moving in a circle at constant speed is constantly changing direction, so its velocity is always changing.

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Representing Vectors with Arrows

Vectors are represented by arrows:

• The length of the arrow represents the magnitude (drawn to scale).
• The direction of the arrow represents the direction of the vector.

For example, a 10 N force to the right and a 5 N force to the left:

graph LR
    A["Start"] -->|"10 N"| B["Right"]
    C["Start"] -->|"5 N"| D["Left"]

    style A fill:#2c3e50,color:#fff
    style B fill:#27ae60,color:#fff
    style C fill:#2c3e50,color:#fff
    style D fill:#c0392b,color:#fff

The longer arrow represents the larger force.

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Adding Vectors Along the Same Line

When two vectors act along the same line, adding them is straightforward:

Same Direction

If two forces act in the same direction, add their magnitudes.

• 10 N right + 5 N right = 15 N right

Opposite Direction

If two forces act in opposite directions, subtract the smaller from the larger. The direction is that of the larger force.

• 10 N right + 5 N left = 5 N right

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Adding Vectors at Right Angles

When two vectors act at right angles to each other, you find the resultant using a scale diagram or Pythagoras' theorem.

Method: Scale Diagram

1. Choose a suitable scale (e.g. 1 cm = 1 N).
2. Draw the first vector as an arrow to scale.
3. Draw the second vector starting from the tip of the first, at 90°.
4. Draw the resultant from the start of the first arrow to the tip of the second.
5. Measure the length and angle of the resultant.

Worked Example

A boat travels 30 m east and then 40 m north. Find the resultant displacement.

Using Pythagoras:

• Resultant² = 30² + 40² = 900 + 1600 = 2500
• Resultant = √2500 = 50 m

Direction:

• tan θ = opposite / adjacent = 40 / 30 = 1.333
• θ = tan⁻¹(1.333) = 53.1° north of east

Exam Tip: When using scale diagrams, always state your scale, use a ruler and protractor, and show your construction lines. The examiner needs to see your working.

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The Parallelogram Rule

For two vectors that are not at right angles, you can use the parallelogram rule:

1. Draw both vectors starting from the same point, to scale.
2. Complete the parallelogram by drawing lines parallel to each vector.
3. The diagonal from the starting point is the resultant vector.
4. Measure the length and angle of the diagonal.

graph TD
    A["Starting Point"] -->|"Vector 1"| B["End of V1"]
    A -->|"Vector 2"| C["End of V2"]
    B -.->|"Parallel to V2"| D["Resultant End"]
    C -.->|"Parallel to V1"| D
    A ==>|"Resultant"| D

    style A fill:#2c3e50,color:#fff
    style B fill:#2980b9,color:#fff
    style C fill:#e67e22,color:#fff
    style D fill:#27ae60,color:#fff

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Summary